Abstract

A function is said to be a Fibonacci function if for all . In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period .

1. Introduction

Presently, there are many research articles about Fibonacci numbers (see [1]). Fibonacci numbers are also involved in the golden ratio (see [2]). In 2008, Kim and Neggers [3] studied Fibonacci means. In 2009, Jung [4] studied Hyers-Ulam stability of Fibonacci functional equation. In 2010, Han et al. [5] studied a Fibonacci norm of positive integers. In 2012, Han et al. [6] studied Fibonacci sequences in groupoids. Moreover, they [7] gave some properties on Fibonacci functions; a function is said to be a Fibonacci function if , for all , using the concept of -even and -odd functions. They also showed that if is a Fibonacci function, then .

In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period using the concept of -even and -odd functions with period . Moreover, we also present some properties on the odd Fibonacci functions with period .

2. Fibonacci Functions with Period

Definition 1. Let be a positive integer. A function is said to be a Fibonacci function with period if for all .

Example 2. Let be a Fibonacci function with period , where . It follows that for all , so . Then . Thus, for all .

Proposition 3. Let be a Fibonacci function with period . Assume that is differentiable. Then is also a Fibonacci function with period .

Proof. Let . Since , it follows that .

Proposition 4. Let be a Fibonacci function with period , and define for all , where . Then is also a Fibonacci function with period .

Proof. Let . Then .

Example 5. Let and . Define by for all . Then is a Fibonacci function with period .

Theorem 6. Let be a Fibonacci function with period , and let be a sequence of Fibonacci numbers with , , and for all . Then, for any and , .

Proof. Let . We note that and . Now, we assume that and , where . Then
This proof is completed.

3. Odd Fibonacci Functions with Period

Definition 7. Let be a positive integer. A function is said to be an odd Fibonacci function with period if for all .

Example 8. Let be an odd Fibonacci function with period , where . It follows that for all , so . Then . Thus, for all .

Proposition 9. Let be an odd Fibonacci function with period . Assume that is differentiable. Then is also an odd Fibonacci function with period .

Proof. Let . Since , it follows that .

Proposition 10. Let be an odd Fibonacci function with period , and define for all , where . Then is also an odd Fibonacci function with period .

Proof. Let . Then .

Example 11. Let and . Define by for all . Then is an odd Fibonacci function with period .

Theorem 12. Let be an odd Fibonacci function with period , and let be a sequence of Fibonucci numbers with , , and for all . Then, for any and , .

Proof. Let . We note that and . Now, we assume that and , where . Then
This proof is completed.

4. -Even Functions with Period

Definition 13. Let be a positive integer and let be such that if , where is continuous, then . The function is said to be an -even function with period if for all .

Example 14. Define for all . Let be a continuous function such that . For any , we have , so . Since is dense in and is continuous, it follows that . Let and . Then . Hence, is an -even function with period .

Theorem 15. Let and be an -even function with period and let be a continuous function. Then is a Fibonacci function with period if and only if is a Fibonacci function with period .

Proof. First, we assume that is a Fibonacci function with period . For any , we have
Hence, is a Fibonacci function with period .Next, we assume that is a Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is a Fibonacci function with period .

Example 16. Let . Define and for all . For all , we have . We recall that is an -even function with period , and is a Fibonacci function with period . Hence, is a Fibonacci function with period .

Theorem 17. Let and be an -even function with period and let be a continuous function. Then is an odd Fibonacci function with period if and only if is an odd Fibonacci function with period .

Proof. First, we assume that is an odd Fibonacci function with period . For any , we have
Hence, is an odd Fibonacci function with period .Next, we assume that is an odd Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is an odd Fibonacci function with period .

Example 18. Let . Define and for all . For all , we have . We recall that is an -even function with period and is an odd Fibonacci function with period . Hence, is an odd Fibonacci function with period .

5. -Odd Functions with Period

Definition 19. Let be a positive integer and let be such that if where is continuous, then . The function is said to be an -odd function with period if for all .

Example 20. Define for all . Let be a continuous function such that . For any , we have , so . Since is dense in and is continuous, it follows that . Let be a positive odd integer and . Then . Hence, is an -even function with period .

Theorem 21. Let and be an -odd function with period and let be a continuous function. Then is a Fibonacci function with period if and only if is an odd Fibonacci function with period .

Proof. First, we assume that is a Fibonacci function with period . For any , we have
Hence, is an odd Fibonacci function with period .Next, we assume that is an odd Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is a Fibonacci function with period .

Example 22. Let be a positive odd integer. Define and for all . We have for all . We recall that is an -odd function with period and is a Fibonacci function with period . Hence, is an odd Fibonacci function with period .

Theorem 23. Let and be an -odd function with period and let be a continuous function. Then is an odd Fibonacci function with period if and only if is a Fibonacci function with period .

Proof. First, we assume that is an odd Fibonacci function with period . For any , we have
Hence, is a Fibonacci function with period .Next, we assume that is a Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is an odd Fibonacci function with period .

Example 24. Let be a positive odd integer. Define and for all . We have for all . We recall that is an -odd function with period and is an odd Fibonacci function with period . Hence, is a Fibonacci function with period .

6. Open Problems

Conjecture 25.
If is a Fibonacci function with period , then

Conjecture 26.
If is an odd Fibonacci function with period , then

Acknowledgments

The author would like to thank the referees for their useful comments and suggestions.